New Method for Estimating Fractal Dimension in 3D Space and Its Application to Complex Surfaces

Matej Babič, George Ch. Miliaresis, Matjaž Mikoš, Rita Ambu, Michele Calì


The concept of “surface modeling” generally describes the process of representing a physical or artificial surface by a geometric model, namely a mathematical expression. Among the existing techniques applied for the characterization of a surface, terrain modeling relates to the representation of the natural surface of the Earth. Cartographic terrain or relief models as three-dimensional representations of a part of the Earth's surface convey an immediate and direct impression of a landscape and are much easier to understand than two-dimensional models. This paper addresses a major problem in complex surface modeling and evaluation consisting in the characterization of their topography and comparison among different textures, which can be relevant in different areas of research. A new algorithm is presented that allows calculating the fractal dimension of images of complex surfaces. The method is used to characterize different surfaces and compare their characteristics. The proposed new mathematical method computes the fractal dimension of the 3D space with the average space component of Hurst exponent H, while the estimated fractal dimension is used to evaluate, compare and characterize complex surfaces that are relevant in different areas of research. Various surfaces with both methods were analyzed and the results were compared. The study confirms that with known coordinates of a surface, it is possible to describe its complex structure. The estimated fractal dimension is proved to be an ideal tool for measuring the complexity of the various surfaces considered.


image analysis; fractal dimension; surface; space component; hurst exponent H.

Full Text:



J. C. Russ, F. B. Neal, The image processing handbook, 17th ed., CRC Press, Francis and Taylor group, Boca Raton, U.S., 2017.

M. Calì, S. M. Oliveri, A. Gloria, M. Martorelli, and D. Speranza. "Comparison of commonly used sail cloths through photogrammetric acquisitions, experimental tests and numerical aerodynamic simulations." Procedia Manufacturing, 11, pp. 1651-1658, 2017.

M. Calì, S. M. Oliveri, U. Cella, M. Martorelli, A. Gloria, & D. Speranza. "Mechanical characterization and modeling of downwind sailcloth in fluid-structure interaction analysis." Ocean Engineering, 165, pp. 488-504, 2018.

F. Beritelli, et al. "Automatic heart activity diagnosis based on Gram polynomials and probabilistic neural networks." Biomedical engineering letters 8.1: pp. 77-85, 2018.

C. Wei, C. Zhanchuan, T. Zesheng. "Fractal structure of lunar topography: An interpretation of topographic characteristics." Geomorphology, Volume 238, 1 pp. 112-118, June 2015.

M. K. Annette. "Critical cartography 2.0: From participatory mapping to authored visualizations of power and people." Landscape and Urban Planning 142 pp. 215–225, 2015.

L. Zhu, Z. He, X. Pann, X. Wu. "An Approach to Computer Modeling of Geological Faults in 3D and an Application." Journal of China University of Mining and Technology. Volume 16, Issue 4, pp. 461–465, December 2006.

A. B. Murray, E. Lazarus, A. Ashton, A. Baas, G. Coco, T. Coulthard, & J. Pelletier. "Geomorphology, complexity, and the emerging science of the Earth's surface." Geomorphology, 103(3), pp. 496-505, 2009.

M. Calì, R. Ambu, "Advanced 3D Photogrammetric Surface Reconstruction of Extensive Objects by UAV Camera Image Acquisition", Sensors 18(9), 2815, Aug.2018.

L. A. Méndez-Barroso, J.L. Zárate-Valdez, A. Robles-Morúa, "Estimation of hydromorphological attributes of a small forested catchment by applying the Structure from Motion (SfM) approach", International Journal of Applied Earth Observation and Geoinformation, vol.69, pp. 186-197, July 2018.

Xu T., I.D. Moore, J.C. Gallant. "Fractal dimensions and landscapes — a review. Geomorphology, 8, pp. 245–262, 1993.

X. H.Shen, L. J.Zou, G. F. Zhang, N. Su, W.Y. Wu, S. F. Yang. "Fractal characteristics of the main channel of Yellow River and its relation to regional tectonic evolution." Geomorphology, Volume 127, Issues 1–2, pp. 64-70, April 2011.

A. C. W. Baas "Chaos, fractals and self-organization in coastal geomorphology: simulating dune landscapes in vegetated environments." Geomorphology, Volume 48, Issues 1–3, pp. 309-328, November 2002.

B. B. Mandelbrot. "The fractal geometry of nature." New York: W. H. Freeman, 93, 1998.

C. Zhiying, L. Yong and Z. Ping, "A comparative study of fractal dimension calculation methods for rough surface profiles." Chaos, Solitons & Fractals, vol.112, pp.24-30, July 2018.

F. Tiago, R. Wayne. ImageJ. User GuidIJ1. 46r. 2002

S. Ouchi, M. Matsushita. "Measurement of self-affinity on surfaces as a trial application of fractal geometry to landform analysis." Geomorphology, Volume 5, Issues 1–2, pp. 115-130, May 1992.

J. Qin, Z. Deyu, W. Guangqian, L. N. Sai. "Influence of particle shape on surface roughness: Dissimilar morphological structures formed by man-made and natural gravels." Geomorphology, Volume 190, pp. 16-26, May 2013.

Q.C. Sung, Y.C. Chen. "Self-affinity dimensions of topography and its implications in morphotectonics: an example from Taiwan." Geomorphology, Volume 62, Issues 3–4, 1, pp. 181-198, Oct. 2004.

M. Babič, P. Kokol, N. Guid, P. Panjan. "A new method for estimating the Hurst exponent H for 3D objects." Materiali in tehnologije 48-2, 2014.

Phillips, J.D.Earth Surface Systems: Complexity, Order and Scale. Blackwell, Oxford, 1999.

Capizzi, Giacomo, et al. "A multithread nested neural network architecture to model surface plasmon polaritons propagation." Micromachines 7.7 (2016): 110.

C. Kumar, A. Palacios, V. A. Surapaneni, G. Bold, M. Thielen, E. Licht, T. E. Higham, T. Speck and V. Le Houérou. "Replicating the complexity of natural surfaces: technique validation and applications for biomimetics, ecology and evolution, 377." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

H. Sun & H. Chen. "Soft-Touch Haptics Modeling of Dynamic Surfaces." Virtual Technologies for Business and Industrial Applications: Innovative and Synergistic Approaches. Editors: N. Raghavendra Rao (VIT University, India), 2011.

C. H. Yan, S. H. Ong, Y. Ge, J. Zhang, S. H. Teoh and B. H. Okker, "A neural network approach for 3D surface modeling and registration," IEEE International Workshop on Biomedical Circuits and Systems, Singapore, pp. S3/2-17, 2004.

I. P. Nanda, M. Hazwan Hassim, M. Hasbullah Idris, M. H. Jahare, A. Arafat. "Effect of Mechanical Tumbling Parameters on Surface Roughness and Edge Radius of Medical Grade Cobalt Chromium Alloy." International Journal on Advanced Science, Engineering and Information Technology, Vol. 9 No. 1, 2019.

S. Brusca, G. Capizzi, G. Lo Sciuto, and G. Susi. "A new design methodology to predict wind farm energy production by means of a spiking neural network based-system." International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 7 2017.

P. Loreti, A. Catini, M. De Luca, L. Bracciale, G. Gentile and C. Di Natale. “The Design of an Energy Harvesting Wireless Sensor Node for Tracking Pink Iguanas”. Sensors, 19(5), 985, 2019.

L. Bracciale, A. Catini, G. Gentile, P. Loreti. "Delay tolerant wireless sensor network for animal monitoring: The Pink Iguana case" Lecture Notes in Electrical Engineering, 429, pp. 18-26, 2017.

R. Zakaria, A. Wahab, & R. U. Gobithaasan. "Fuzzy B-Spline surface modeling." Journal of Applied Mathematics, 2014.

A. Gálvez and A. Iglesias. "Firefly Algorithm for Polynomial Bézier Surface Parameterization." Journal of applied mathematics, 2013.

U. Reuter, A. Sultan and D. Reischl. "A comparative study of machine learning approaches for modeling concrete failure surfaces." Advances in Engineering Software Volume 116, pp. 67-79, February 2018.

C. T. Yeu, M. Lim and G. Huang, "Terrain Modeling Using Machine Learning Methods," 9th International Conference on Control, Automation, Robotics and Vision, pp. 1-4, Singapore, 2006.

S. Angra and S. Ahuja, "Machine learning and its applications: A review,"IEEE International Conference on Big Data Analytics and Computational Intelligence (ICBDAC), pp. 57-60, Chirala, 2017.

Cardarilli G.C. et al. "Efficient Ensemble Machine Learning Implementation on FPGA Using Partial Reconfiguration," Applications in Electronics Pervading Industry, Environment and Society. ApplePies 2018. Lecture Notes in Electrical Engineering, vol 550. Springer, Cham.

G. Susi, L. Anton Toro, L. Canuet, M. E. Lopez, F. Maestu, C. R. Mirasso, and E. Pereda, “A neuro-inspired system for online learning and recognition of parallel spike trains, based on spike latency, and heterosynaptic STDP”. in Frontiers in Neuroscience, 12:780, 2018. ISSN 1662-453X. doi: 10.3389/fnins.2018.00780.

M. Elangovan, N. R. Sakthivel, S. Saravanamurugan, Binoy. B. Nair, V. Sugumaran. "Machine Learning Approach to the Prediction of Surface Roughness Using Statistical Features of Vibration Signal Acquired in Turning." Procedia computer science, 2015.

D. Racki, D. Tomazevic, D. Skocaj. "A Compact Convolutional Neural Network for Textured Surface Anomaly Detection" IEEE Winter Conference on Applications of Computer Vision (WACV). 2018.

M. Matta, G. C. Cardarilli, L. Di Nunzio, R. Fazzolari, D. Giardino, M. Re, F. Silvestri, S. Spanò, "Q-RTS: a real-time swarm intelligence based on multi-agent Q-learning." Electronics Letters, 55 (10), pp. 589-591, 2019.

G. C. Cardarilli, L. Di Nunzio, R. Fazzolari, D. Giardino, M. Matta, M. Re, F. Silvestri and S. Spanò "Efficient Ensemble Machine Learning implementation on FPGA using Partial Reconfiguration." Lecture Notes in Electrical Engineering. 2019 ARTICLE IN PRESS.

G. C. Cardarilli, L. Di Nunzio, R. Fazzolari, M. Re, S. Spanó "AW-SOM, an Algorithm for High-speed Learning in Hardware Self-Organizing Maps" IEEE Transactions on Circuits and Systems II: Express Briefs DOI:10.1109/TCSII.2019.2909117, 2019 ARTICLE IN PRESS.



  • There are currently no refbacks.

Published by INSIGHT - Indonesian Society for Knowledge and Human Development